
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a field is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
on which
addition,
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
, and
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
* Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
are defined and behave as the corresponding operations on
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
and
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s do. A field is thus a fundamental
algebraic structure which is widely used in
algebra,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, and many other areas of mathematics.
The best known fields are the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
s, the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s and the field of
complex numbers. Many other fields, such as
fields of rational functions,
algebraic function fields,
algebraic number fields, and
''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. Most
cryptographic protocol
A security protocol (cryptographic protocol or encryption protocol) is an abstract or concrete protocol that performs a security-related function and applies cryptographic methods, often as sequences of cryptographic primitives. A protocol descri ...
s rely on
finite fields, i.e., fields with finitely many
elements.
The relation of two fields is expressed by the notion of a
field extension.
Galois theory, initiated by
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that
angle trisection and
squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficul ...
cannot be done with a
compass and straightedge. Moreover, it shows that
quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
s are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the
scalars for a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, which is the standard general context for
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
.
Number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
s, the siblings of the field of rational numbers, are studied in depth in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
.
Function fields can help describe properties of geometric objects.
Definition
Informally, a field is a set, along with two
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
s defined on that set: an addition operation written as , and a multiplication operation written as , both of which behave similarly as they behave for
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
s and
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s, including the existence of an
additive inverse for all elements , and of a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
for every nonzero element . This allows one to also consider the so-called ''inverse'' operations of subtraction, , and division, , by defining:
:,
:.
Classic definition
Formally, a field is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
together with two
binary operations on called ''addition'' and ''multiplication''. A binary operation on is a mapping , that is, a correspondence that associates with each ordered pair of elements of a uniquely determined element of . The result of the addition of and is called the sum of and , and is denoted . Similarly, the result of the multiplication of and is called the product of and , and is denoted or . These operations are required to satisfy the following properties, referred to as ''
field axioms'' (in these axioms, , , and are arbitrary
element
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of o ...
s of the field ):
*
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
of addition and multiplication: , and .
*
Commutativity of addition and multiplication: , and .
*
Additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with ...
and
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
: there exist two different elements and in such that and .
*
Additive inverses: for every in , there exists an element in , denoted , called the ''additive inverse'' of , such that .
*
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
s: for every in , there exists an element in , denoted by or , called the ''multiplicative inverse'' of , such that .
*
Distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
of multiplication over addition: .
This may be summarized by saying: a field has two operations, called addition and multiplication; it is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.
Even more summarized: a field is a
commutative ring where
and all nonzero elements are
invertible under multiplication.
Alternative definition
Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.
Division by zero
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there ...
is, by definition, excluded. In order to avoid
existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two
nullary operations (the constants and ). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in
constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...
and
computing. One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two constants and , since and .
[The a priori twofold use of the symbol "−" for denoting one part of a constant and for the additive inverses is justified by this latter condition.]
Examples
Rational numbers
Rational numbers have been widely used a long time before the elaboration of the concept of field.
They are numbers that can be written as
fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, where and are
integers, and . The additive inverse of such a fraction is , and the multiplicative inverse (provided that ) is , which can be seen as follows:
:
The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:
:
Real and complex numbers

The
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s , with the usual operations of addition and multiplication, also form a field. The
complex numbers consist of expressions
: with real,
where is the
imaginary unit, i.e., a (non-real) number satisfying .
Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for . For example, the distributive law enforces
:
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the
plane, with
Cartesian coordinates given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.
Constructible numbers

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with
compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of
constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only
compass and
straightedge
A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler.
Straightedges are used in the automotive service and ma ...
. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
s of constructible numbers, not necessarily contained within . Using the labeling in the illustration, construct the segments , , and a
semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
over (center at the
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dim ...
), which intersects the
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
line through in a point , at a distance of exactly
from when has length one.
Not all real numbers are constructible. It can be shown that
is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a
cube with volume 2, another problem posed by the ancient Greeks.
A field with four elements
In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called , , , and . The notation is chosen such that plays the role of the additive identity element (denoted 0 in the axioms above), and is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
: , which equals , as required by the distributivity.
This field is called a
finite field with four elements, and is denoted or . The
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
consisting of and (highlighted in red in the tables at the right) is also a field, known as the ''
binary field'' or . In the context of
computer science and
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
, and are often denoted respectively by ''false'' and ''true'', and the addition is then denoted
XOR (exclusive or). In other words, the structure of the binary field is the basic structure that allows computing with
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...
s.
Elementary notions
In this section, denotes an arbitrary field and and are arbitrary
elements of .
Consequences of the definition
One has and . In particular, one may deduce the additive inverse of every element as soon as one knows .
If then or must be 0, since, if , then
. This means that every field is an
integral domain.
In addition, the following properties are true for any elements and :
:
:
:
:
: if
The additive and the multiplicative group of a field
The axioms of a field imply that it is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
under addition. This group is called the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structur ...
of the field, and is sometimes denoted by when denoting it simply as could be confusing.
Similarly, the ''nonzero'' elements of form an abelian group under multiplication, called the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred ...
, and denoted by or just or .
A field may thus be defined as set equipped with two operations denoted as an addition and a multiplication such that is an abelian group under addition, is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is
distributive over addition.
[Equivalently, a field is an algebraic structure of type , such that is not defined, and
are abelian groups, and ⋅ is distributive over +. ] Some elementary statements about fields can therefore be obtained by applying general facts of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. For example, the additive and multiplicative inverses and are uniquely determined by .
The requirement follows, because 1 is the identity element of a group that does not contain 0. Thus, the
trivial ring, consisting of a single element, is not a field.
Every finite
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
of the multiplicative group of a field is
cyclic (see ).
Characteristic
In addition to the multiplication of two elements of ''F'', it is possible to define the product of an arbitrary element of by a positive
integer to be the -fold sum
: (which is an element of .)
If there is no positive integer such that
:,
then is said to have
characteristic 0. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there ''is'' a positive integer satisfying this equation, the smallest such positive integer can be shown to be a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. It is usually denoted by and the field is said to have characteristic then.
For example, the field has characteristic 2 since (in the notation of the above addition table) .
If has characteristic , then for all in . This implies that
:,
since all other
binomial coefficients appearing in the
binomial formula
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
are divisible by . Here, ( factors) is the -th power, i.e., the -fold product of the element . Therefore, the
Frobenius map
:
is compatible with the addition in (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic quite different from fields of characteristic 0.
Subfields and prime fields
A ''
subfield'' of a field is a subset of that is a field with respect to the field operations of . Equivalently is a subset of that contains , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that , that for all both and are in , and that for all in , both and are in .
Field homomorphisms are maps between two fields such that , , and , where and are arbitrary elements of . All field homomorphisms are
injective. If is also
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
, it is called an
isomorphism (or the fields and are called isomorphic).
A field is called a
prime field
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive i ...
if it has no proper (i.e., strictly smaller) subfields. Any field contains a prime field. If the characteristic of is (a prime number), the prime field is isomorphic to the finite field introduced below. Otherwise the prime field is isomorphic to .
Finite fields
''Finite fields'' (also called ''Galois fields'') are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements .

The simplest finite fields, with prime order, are most directly accessible using
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his b ...
. For a fixed positive integer , arithmetic "modulo " means to work with the numbers
:
The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. For example, taking the prime results in the above-mentioned field . For and more generally, for any
composite number (i.e., any number which can be expressed as a product of two strictly smaller natural numbers), is not a field: the product of two non-zero elements is zero since in , which, as was explained
above, prevents from being a field. The field with elements ( being prime) constructed in this way is usually denoted by .
Every finite field has elements, where is prime and . This statement holds since may be viewed as a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over its prime field. The
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
of this vector space is necessarily finite, say , which implies the asserted statement.
A field with elements can be constructed as the
splitting field of the
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
:.
Such a splitting field is an extension of in which the polynomial has zeros. This means has as many zeros as possible since the
degree of is . For , it can be checked case by case using the above multiplication table that all four elements of satisfy the equation , so they are zeros of . By contrast, in , has only two zeros (namely 0 and 1), so does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of ''the'' finite field with elements, denoted by or .
History
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations,
algebraic number theory, and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. A first step towards the notion of a field was made in 1770 by
Joseph-Louis Lagrange, who observed that permuting the zeros of a
cubic polynomial
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
in the expression
:
(with being a third
root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
) only yields two values. This way, Lagrange conceptually explained the classical solution method of
Scipione del Ferro and
François Viète, which proceeds by reducing a cubic equation for an unknown to a quadratic equation for . Together with a similar observation for
equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups.
Vandermonde, also in 1770, and to a fuller extent,
Carl Friedrich Gauss, in his ''
Disquisitiones Arithmeticae'' (1801), studied the equation
:
for a prime and, again using modern language, the resulting cyclic
Galois group. Gauss deduced that a
regular -gon can be constructed if . Building on Lagrange's work,
Paolo Ruffini claimed (1799) that
quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
s (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
in 1824.
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as
Galois theory today. Both Abel and Galois worked with what is today called an
algebraic number field, but conceived neither an explicit notion of a field, nor of a group.
In 1871
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by .
In 1881
Leopold Kronecker defined what he called a ''domain of rationality'', which is a field of
rational fractions in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as abstractly as the rational function field . Prior to this, examples of transcendental numbers were known since
Joseph Liouville's work in 1844, until
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
(1873) and
Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficie ...
(1882) proved the transcendence of and , respectively.
The first clear definition of an abstract field is due to . In particular,
Heinrich Martin Weber's notion included the field F
''p''.
Giuseppe Veronese (1891) studied the field of formal power series, which led to introduce the field of ''p''-adic numbers. synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections
Galois theory,
Constructing fields and
Elementary notions can be found in Steinitz's work. linked the notion of
orderings in a field, and thus the area of analysis, to purely algebraic properties.
Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extens ...
.
Constructing fields
Constructing fields from rings
A
commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses . For example, the integers form a commutative ring, but not a field: the
reciprocal of an integer is not itself an integer, unless .
In the hierarchy of algebraic structures fields can be characterized as the commutative rings in which every nonzero element is a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
(which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
s, and . Fields are also precisely the commutative rings in which is the only
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wit ...
.
Given a commutative ring , there are two ways to construct a field related to , i.e., two ways of modifying such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of is , the rationals, while the residue fields of are the finite fields .
Field of fractions
Given an
integral domain , its
field of fractions is built with the fractions of two elements of exactly as Q is constructed from the integers. More precisely, the elements of are the fractions where and are in , and . Two fractions and are equal if and only if . The operation on the fractions work exactly as for rational numbers. For example,
:
It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.
The field of the
rational fractions over a field (or an integral domain) is the field of fractions of the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
. The field of
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...
:
over a field is the field of fractions of the ring of
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
(in which ). Since any Laurent series is a fraction of a power series divided by a power of (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.
Residue fields
In addition to the field of fractions, which embeds
injectively into a field, a field can be obtained from a commutative ring by means of a
surjective map onto a field . Any field obtained in this way is a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
, where is a
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
of . If
has only one maximal ideal , this field is called the
residue field of .
The
ideal generated by a single polynomial in the polynomial ring (over a field ) is maximal if and only if is
irreducible in , i.e., if cannot be expressed as the product of two polynomials in of smaller
degree. This yields a field
:
This field contains an element (namely the
residue class of ) which satisfies the equation
:.
For example, is obtained from by
adjoining the
imaginary unit symbol , which satisfies , where . Moreover, is irreducible over , which implies that the map that sends a polynomial to yields an isomorphism
:
Constructing fields within a bigger field
Fields can be constructed inside a given bigger container field. Suppose given a field , and a field containing as a subfield. For any element of , there is a smallest subfield of containing and , called the subfield of ''F'' generated by and denoted . The passage from to is referred to by ''
adjoining an element'' to . More generally, for a subset , there is a minimal subfield of containing and , denoted by .
The
compositum of two subfields and of some field is the smallest subfield of containing both and The compositum can be used to construct the biggest subfield of satisfying a certain property, for example the biggest subfield of , which is, in the language introduced below, algebraic over .
[Further examples include the maximal unramified extension or the maximal abelian extension within .]
Field extensions
The notion of a subfield can also be regarded from the opposite point of view, by referring to being a ''
field extension'' (or just extension) of , denoted by
:,
and read " over ".
A basic datum of a field extension is its
degree , i.e., the dimension of as an -vector space. It satisfies the formula
:.
Extensions whose degree is finite are referred to as finite extensions. The extensions and are of degree 2, whereas is an infinite extension.
Algebraic extensions
A pivotal notion in the study of field extensions are
algebraic elements. An element is ''algebraic'' over if it is a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
with
coefficients in , that is, if it satisfies a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
:,
with in , and .
For example, the
imaginary unit in is algebraic over , and even over , since it satisfies the equation
:.
A field extension in which every element of is algebraic over is called an
algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.
The subfield generated by an element , as above, is an algebraic extension of if and only if is an algebraic element. That is to say, if is algebraic, all other elements of are necessarily algebraic as well. Moreover, the degree of the extension , i.e., the dimension of as an -vector space, equals the minimal degree such that there is a polynomial equation involving , as above. If this degree is , then the elements of have the form
:
For example, the field of
Gaussian rationals is the subfield of consisting of all numbers of the form where both and are rational numbers: summands of the form (and similarly for higher exponents) don't have to be considered here, since can be simplified to .
Transcendence bases
The above-mentioned field of
rational fractions , where is an
indeterminate, is not an algebraic extension of since there is no polynomial equation with coefficients in whose zero is . Elements, such as , which are not algebraic are called
transcendental. Informally speaking, the indeterminate and its powers do not interact with elements of . A similar construction can be carried out with a set of indeterminates, instead of just one.
Once again, the field extension discussed above is a key example: if is not algebraic (i.e., is not a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of a polynomial with coefficients in ), then is isomorphic to . This isomorphism is obtained by substituting to in rational fractions.
A subset of a field is a
transcendence basis if it is
algebraically independent (don't satisfy any polynomial relations) over and if is an algebraic extension of . Any field extension has a transcendence basis. Thus, field extensions can be split into ones of the form (
purely transcendental extensions) and algebraic extensions.
Closure operations
A field is
algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
:, with coefficients ,
has a solution . By the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, is algebraically closed, i.e., ''any'' polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are ''not'' algebraically closed since the equation
:
does not have any rational or real solution. A field containing is called an ''
algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansk ...
'' of if it is
algebraic over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations).
By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the
Artin-Schreier theorem, the degree of this extension is necessarily 2, and is
elementarily equivalent to . Such fields are also known as
real closed fields.
Any field has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as ''the'' algebraic closure and denoted . For example, the algebraic closure of is called the field of
algebraic numbers. The field is usually rather implicit since its construction requires the
ultrafilter lemma
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X ( ...
, a set-theoretic axiom that is weaker than the
axiom of choice. In this regard, the algebraic closure of , is exceptionally simple. It is the union of the finite fields containing (the ones of order ). For any algebraically closed field of characteristic 0, the algebraic closure of the field of
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...
is the field of
Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
: \begin
x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\
&=x^+ 2x^ + x^ + 2x^ + x^ + ...
, obtained by adjoining roots of .
Fields with additional structure
Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.
Ordered fields
A field ''F'' is called an ''ordered field'' if any two elements can be compared, so that and whenever and . For example, the real numbers form an ordered field, with the usual ordering . The
Artin-Schreier theorem states that a field can be ordered if and only if it is a
formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above i ...
, which means that any quadratic equation
:
only has the solution . The set of all possible orders on a fixed field is isomorphic to the set of
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
s from the
Witt ring of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s over , to .
An
Archimedean field is an ordered field such that for each element there exists a finite expression
:
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no
infinitesimals
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally ref ...
(elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of .

An ordered field is
Dedekind-complete
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...
if all
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an elem ...
s,
lower bounds (see
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rat ...
) and limits, which should exist, do exist. More formally, each
bounded subset of is required to have a least upper bound. Any complete field is necessarily Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence , every element of which is greater than every infinitesimal, has no limit.
Since every proper subfield of the reals also contains such gaps, is the unique complete ordered field, up to isomorphism. Several foundational results in
calculus follow directly from this characterization of the reals.
The
hyperreals
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of
non-standard analysis.
Topological fields
Another refinement of the notion of a field is a
topological field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
, in which the set is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
, such that all operations of the field (addition, multiplication, the maps and ) are
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
s with respect to the topology of the space.
The topology of all the fields discussed below is induced from a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
, i.e., a
function
:
that measures a ''distance'' between any two elements of .
The
completion of is another field in which, informally speaking, the "gaps" in the original field are filled, if there are any. For example, any
irrational number , such as , is a "gap" in the rationals in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers , in the sense that distance of and given by the
absolute value is as small as desired.
The following table lists some examples of this construction. The fourth column shows an example of a zero
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
, i.e., a sequence whose limit (for ) is zero.
The field is used in number theory and
-adic analysis. The algebraic closure carries a unique norm extending the one on , but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of
complex p-adic numbers and is denoted by .
Local fields
The following topological fields are called ''
local fields'':
[Some authors also consider the fields and to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that calls them "completely anomalous".]
* finite extensions of (local fields of characteristic zero)
* finite extensions of , the field of Laurent series over (local fields of characteristic ).
These two types of local fields share some fundamental similarities. In this relation, the elements and (referred to as
uniformizer) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in . (However, since the addition in is done using
carrying, which is not the case in , these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper:
* Any
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
statement that is true for almost all is also true for almost all . An application of this is the
Ax-Kochen theorem describing zeros of homogeneous polynomials in .
*
Tamely ramified extensions of both fields are in bijection to one another.
* Adjoining arbitrary -power roots of (in ), respectively of (in ), yields (infinite) extensions of these fields known as
perfectoid fields. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields:
Differential fields
Differential fields are fields equipped with a
derivation, i.e., allow to take derivatives of elements in the field. For example, the field R(''X''), together with the standard derivative of polynomials forms a differential field. These fields are central to
differential Galois theory, a variant of Galois theory dealing with
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
s.
Galois theory
Galois theory studies
algebraic extensions of a field by studying the
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
in the arithmetic operations of addition and multiplication. An important notion in this area is that of
finite Galois extensions , which are, by definition, those that are
separable and
normal. The
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extens ...
shows that finite separable extensions are necessarily
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, i.e., of the form
:,
where is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of are contained in and that has only simple zeros. The latter condition is always satisfied if has characteristic 0.
For a finite Galois extension, the
Galois group is the group of
field automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of that are trivial on (i.e., the
bijections that preserve addition and multiplication and that send elements of to themselves). The importance of this group stems from the
fundamental theorem of Galois theory, which constructs an explicit
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the set of
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
s of and the set of intermediate extensions of the extension . By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not
solvable (cannot be built from
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s), then the zeros of ''cannot'' be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving